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Question - Math 324 Assignment 6 (due October 29)

1. (7) If m ≥ 1 is an odd integer with exactly s di stinct prime factors, determine the
integer N of Problem 4.7. Hint: Theorem 4.13 and Problem 4.4.

2. ( 6) If r is a primitive root for a prime p ≡ 1 mod 4, show that − r is also a primitive root
mod p.

3. ( 3 + 6) Let q be an odd prime. Show that
a) there are exactly two real-value d Dirichlet characters mod q, and
b) since the principal character χ0 mod q is one of the two above, let χ1 be the other one
and show that, for all integers b with (b, q) = 1, we have
(i) χ1(b) = 1, if x2 ≡ b mod q has a solution x, and
(ii) χ1(b) = − 1, if x2 ≡ b mod q has no solution mod q.

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Solution Preview - lutions x mod m of x2 ≡ 1 mod m. Let m = p1a(1) … psa(s) be the prime power factorization of m, with a(i) ≥ 1 for all i. Each solution x of x2 ≡ 1 mod m is a simultaneous solution of the equations x2 ≡ 1 mod pia(i) for 1 ≤ i ≤ s. For each i, this equation mod pia(i) has exactly 2 solutions x ≡ ± 1 pia(i), by Problem 4.4, so each

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